\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 148, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/148\hfil A remark on regularity]
{A remark on the regularity for the 3D Navier-Stokes equations
in terms of the two components of the velocity}

\author[S. Gala\hfil EJDE-2009/148\hfilneg]
{Sadek Gala}

\address{Sadek Gala \newline
 Department of Mathematics, University of Mostaganem \\
 Box 227, Mostaganem 27000, Algeria}
\email{sadek.gala@gmail.com}

\thanks{Submitted November 5, 2009. Published November 25, 2009.}
\subjclass[2000]{35Q30, 35K15, 76D05}
\keywords{Navier-Stokes equations; regularity criterion;
\hfill\break\indent  Morrey-Campanato spaces}

\begin{abstract}
 In this note, we study the regularity of Leray-Hopf weak solutions
 to the Navier-Stokes equation, with the condition
 \[
 \nabla (u_{1},u_{2},0)
 \in L^{\frac{2}{1-r}}(0,T; \dot{\mathcal{M}}_{2,3/r} (\mathbb{R}^3) ,
 \]
 where $\dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ is the
 Morrey-Campanato space for $0<r<1$. Since
 \[
 L^{1/3}(\mathbb{R}^3)\subset \dot{X}_r(\mathbb{R}^3) 
 \subset \dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3),
 \]
 the above regularity condition allows us to improve the results
 obtained by Fan and Gao \cite{FG}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

Consider the Navier-Stokes equation, in $\mathbb{R}^3$,
\begin{equation}  \label{eqNS}
\begin{gathered}
\partial _{t}u+(u\cdot \nabla )u-\Delta u+\nabla p =0,\quad
(x,t)\in \mathbb{R}^3\times (0,T),   \\
\mathop{\rm div}u = 0,\quad (x,t)\in \mathbb{R}^3\times (0,T),\\
u(x,0) = u_0(x),\quad x\in \mathbb{R}^3,
\end{gathered}
\end{equation}
where $u=u(x,t)$ is the velocity field, $p=p(x,t)$ is the scalar pressure
and $u_0(x)$ with $\mathop{\rm div }u_0=0$ in the sense of
distribution is the initial velocity field. For simplicity,
we assume that the external force
has a scalar potential and is included in the pressure gradient.

In their classical article, Leray \cite{Ler} and Hopf \cite{Hop}
independently constructed a weak solution $u$ of \eqref{eqNS} for
arbitrary $u_0\in L^{2}( \mathbb{R}^3) $ with $\mathop{\rm div }u_0=0$.
The solution is called the Leray-Hopf weak solution.
Regularity of such Leray-Hopf weak solutions is
one of the most significant open problems in mathematical fluid mechanics.

By a weak solution we mean a function
$u\in L^{\infty } ( 0,T ;L^{2}(\mathbb{R}^3) )
\cap L^{2}( 0,T;\dot{H}^{1}( \mathbb{R}^3 )) $
satisfying \eqref{eqNS} in sense of distributions.
See e.g. \cite{Soh} for an exposition of the theory of weak solutions.

Introducing the class $L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))$, it is shown
that if we have a Leray-Hopf weak solution $u$ belonging to
$L^{\alpha }((0,T);L^{q}(\mathbb{R}^3))$ with the exponents $\alpha $
and $q$ satisfying $\frac{2}{\alpha }+\frac{3}{q}\leq 1$,
$2\leq \alpha <\infty $, $3<q\leq \infty $, then the solution
$u(x,t)\in C^{\infty }(\mathbb{R}^3\times (0,T))$
\cite{Ser, ohy, pro, fab, G, Zhou1, Zhou2}. The
limit case $\alpha =\infty $, $q=3$ was covered much
later Escauriaza, Seregin and Sverak in \cite{sve}.
Bae and Choe \cite{BC} proved that $u$ is strong if
$\widetilde{u}\in L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))$ with
 $\frac{2}{\alpha }+\frac{3}{q}=1$ and $q>3$.
Later, Chae-Choe \cite{DC} obtained an improved regularity
criterion of \cite{B} imposing condition on only two components of the
velocity, namely if
\begin{gather*}
\nabla \tilde u \in L^{\alpha }(0,T;L^{q}(\mathbb{R}^3))\quad
\text{with }\frac{2}{\alpha }+\frac{3}{q}\leq 2,\;
1\leq\alpha <\infty,\\
\tilde u =(u_{1},u_{2},0)
\end{gather*}
then the weak solution becomes smooth. See also \cite{ZP1, ZP2}
for recent improvements of these criteria, via one velocity component.
Recently, Fan and Gao \cite{FG} improved the regularity criterion
in \cite{DC}, under the condition
\[
\nabla \tilde u  \in L^{\frac{2}{2-r}}(0,T;
\dot{X}_r(\mathbb{R}^3)\mathbb{)}\quad \text{for some }0<r<1,
\]
where $\dot{X}_r$ is the multiplier space (see definition below).

The purpose of this note is to imporve the results in \cite{DC}
and \cite{FG}, by proving that if
$\nabla \tilde u \in L^{\frac{2}{2-r}}
(0,T;\mathcal{\dot{M}}_{2,3/r}(\mathbb{R}^3))$ with $0<r<1$,
then the weak solution becomes smooth.
Here $\mathcal{\dot{M}}_{2,3/r}(\mathbb{R}^3) $ is the Morrey-Campanato
space, which is strictly larger than
$L^{1/3}( \mathbb{R}^3) $ and $\dot{X}_r(\mathbb{R}^3)$
(see the next section for the related embedding relations).

\section{Preliminaries and the main result}

Now, we recall the definition and some properties of the spaces to be
used later. These spaces play an important role in studying
the regularity of solutions to partial differential equations;
see e.g. \cite{GL} and the references therein.

\begin{definition} \label{def2.1} \rm
For $0\leq r<3/2$, the space $\dot{X}_r(\mathbb{R}^3)$ is
defined as the space of functions $f(x)\in L_{\rm loc}^{2}(\mathbb{R}^3) $
such that
\[
\| f\| _{\dot{X}_r}=\sup_{\|g\| _{\dot{H}^{r}}\leq 1} \| fg\|
_{L^{2}}<\infty .
\]
where we denote by $\dot{H}^{r}(\mathbb{R}^3)$ the
completion of the space $C_0^{\infty }(\mathbb{R}^3)$ with
respect to the norm
$\| u\| _{\dot{H}^{r}}=\| (-\Delta )^{r/2}u\| _{L^{2}}$.
\end{definition}

We have the following homogeneity properties:
For all $x_0\in \mathbb{R}^3$,
\begin{gather*}
\| f(\cdot +x_0)\| _{\dot{X}_r} =\|f\| _{\dot{X}_r} \\
\| f(\lambda \cdot )\| _{\dot{X}_r}
=\frac{1}{\lambda ^{r}}\| f\| _{\dot{X}_r}, \quad \lambda >0.
\end{gather*}
Also we have the imbedding
\[
L^{1/3}(\mathbb{R}^3) \hookrightarrow \dot{X}_r(\mathbb{R}^3)
\quad \text{for }0\leq r<\frac{3}{2}\,.
\]
Now we recall the definition of the Morrey-Campanato spaces.

\begin{definition} \label{def2.2} \rm
For $1<p\leq q\leq +\infty $, the Morrey-Campanato space
$\dot{\mathcal{M}}_{p,q}(\mathbb{R}^3)$ is defined by
\begin{equation}
\dot{\mathcal{M}}_{p,q}(\mathbb{R}^3)=\big\{ f\in
L_{\rm loc}^{p}(\mathbb{R}^3): \|f\| _{\dot{\mathcal{M}}_{p,q}}
=\sup_{x\in \mathbb{R}^3} \sup_{R>0} R^{3/q-3/p}\|f\| _{L^{p}(B(x,R))}
<\infty \}  \label{eq1.13}
\end{equation}
\end{definition}

It is easy to check the equality
\[
\| f(\lambda \cdot)\| _{\dot{\mathcal{M}}
_{p,q}}=\frac{1}{\lambda ^{3/q}} \| f\| _{\dot{\mathcal{M}}_{p,q}},
\quad \lambda >0.
\]
For $2<p\leq 3/r$ and $0<r<3/2$ we have the following
embeddings:
\[
L^{1/3}(\mathbb{R}^3)\hookrightarrow L^{3/r,\infty }
(\mathbb{R}^3)\hookrightarrow \dot{\mathcal{M}}_{p,3/r}
(\mathbb{R}^3)\hookrightarrow \dot{X}_r(\mathbb{R}^3)\hookrightarrow
\mathcal{\dot{M}}_{2,3/r}(\mathbb{R}^3).
\]
The relation
\[
L^{3/r,\infty }(\mathbb{R}^3)\hookrightarrow
\dot{\mathcal{M}}_{p,3/r}(\mathbb{R}^3)
\]
is shown as follows.
\begin{align*}
\| f\| _{\dot{\mathcal{M}}_{p,\frac{3}{r}}}
&\leq \sup_{E}| E| ^{\frac{r}{3}-\frac{1}{2}}\Big(
\int_E | f(y)| ^{p}dy\Big)^{1/p}\quad (f\in L^{3/r,\infty }(\mathbb{R}^3))\\
&=\Big(\sup_E| E| ^{\frac{pr}{3}-1}\int_{E} | f(y)| ^{p}dy\Big)^{1/p}
  \\
&\cong \Big(\sup_{R>0} R|\{ x\in \mathbb{R}^3:| f(y)| ^{p}>R\} |
 ^{pr/3}\Big)^{1/p} \\
&=\sup_{R>0}  R| \{ x\in \mathbb{R}^{p}:|f(y)| >R\} | ^{r/3} \\
&\cong \| f\| _{L^{3/r,\infty }}.
\end{align*}
For $0<r<1$, we use the fact that
\[
L^{2} \cap \dot{H}^{1}\subset \dot{B}_{2,1}^{r}\subset \dot{H}^{r}.
\]
Thus we can replace the space $\dot{X}_r$ by the pointwise
multipliers from Besov space $\dot{B}_{2,1}^{r}$ to $L^{2}$. Then we
have the following lemma given in \cite{Lem1}.

\begin{lemma} \label{lem2}
For $0\leq r<3/2$, the space $\dot{Z}_r(\mathbb{R}^3)$ is defined
as the space of functions
$f(x)\in L_{\rm loc}^{2}(\mathbb{R}^3)$ such that
\[
\| f\| _{\dot{Z}_r}=\sup_{\|g\| _{\dot{B}_{2,1}^{r}}\leq 1}
\|fg\| _{L^{2}}<\infty .
\]
Then $f\in \dot{\mathcal{M}}_{2,3/r}(\mathbb{R}^3)$ if and only
if $f\in \dot{Z}_r(\mathbb{R}^3)$ with equivalence of norms.
\end{lemma}

Additionally, for $2<p\leq \frac{3}{r}$ and $0\leq r<\frac{3}{2}$, we have
the following inclusions \cite{Lem,Lem1}:
\[
\dot{\mathcal{M}}_{p,3/r}(\mathbb{R}^3)
\hookrightarrow \dot{X}_r(\mathbb{R}^3)
\hookrightarrow\dot{\mathcal{M}}_{2,3/r}(\mathbb{R}
^3)=\dot{Z}_r(\mathbb{R}^3)\text{.}
\]
The relation
\[
\dot{X}_r(\mathbb{R}^3)\hookrightarrow \dot{
\mathcal{M}}_{2,3/r}(\mathbb{R}^3)
\]
is shown as follows: Let $f\in \dot{X}_r(\mathbb{R}
^3)$, $0<R\leq 1$, $x_0\in \mathbb{R}^3$ and
$\phi \in C_0^{\infty }(\mathbb{R}^3)$, $\phi \equiv 1$ on
$B(\frac{x_0}{R},1)$. We have
\begin{align*}
R^{r-\frac{3}{2}}\Big(\int_{| x-x_0| \leq
R}| f(x)| ^{2}dx\Big)^{1/2}
& =R^{r}\Big(\int_{| y-\frac{x_0}{R}| \leq 1}| f(Ry)| ^{2}dy\Big)^{1/2} \\
& \leq R^{r}\Big(\int_{y\in \mathbb{R}^3}| f(Ry)\phi
 (y)| ^{2}dy\Big)^{1/2} \\
& \leq R^{r}\| f(R.)\| _{\dot{X}_r}\| \phi \| _{H^{r}} \\
& \leq \| f\| _{\dot{X}_r}\| \phi \| _{H^{r}} \\
& \leq C\| f\| _{\dot{X}_r}.
\end{align*}

The following result well be used in the proof of
Theorem \ref{th1}.

\begin{lemma} \label{lem3}
For $0<r<1$, we have
\[
\| f\| _{\dot{B}_{2,1}^{r}}\leq C\|f\| _{L^{2}}^{1-r}\|
\nabla f\| _{L^{2}}^{r}.
\]
\end{lemma}

\begin{proof}
The idea comes from \cite{MO} (see also \cite{ZG}). According to the
definition of Besov spaces, one has
\begin{align*}
\| f\| _{\dot{B}_{2,1}^{r}}
&= \sum_{j\in \mathbb{Z}}2^{jr}\| \Delta _{j}f\| _{L^{2}} \\
&\leq \sum_{j\leq k}2^{jr}\| \Delta _{j}f\|
_{L^{2}}+\sum_{j>k}2^{j(r-1)}2^{j}\| \Delta _{j}f\|
_{L^{2}} \\
&\leq (\sum_{j\leq k}2^{2jr})^{1/2}(
\sum_{j\leq k}\| \Delta _{j}f\| _{L^{2}}^{2})
^{1/2}+(\sum_{j>k}2^{2j(r-1)})^{\frac{1}{2}
}(\sum_{j>k}2^{2j}\| \Delta _{j}f\|
_{L^{2}}^{2})^{1/2} \\
&\leq C\Big(2^{rk}\| f\| _{L^{2}}+2^{k(r-1)}\|
f\| _{\dot{H}^{1}}\Big)\\
&= C(2^{rk}A^{-r}+2^{k(r-1)}A^{1-r})\| f\|
_{L^{2}}^{1-r}\| f\| _{\dot{H}^{1}}^{r},
\end{align*}
where $A=\| f\| _{\dot{H}^{1}}/ \|f\| _{L^{2}}$.

Choose $k$ such that $2^{rk}A^{-r}\leq 1$; that
is, $k\leq [ \log A^{r}] $. Then
\begin{align*}
\| f\| _{\dot{B}_{2,1}^{r}}
&\leq C( 1+2^{k(r-1)}A^{1-r})\| f\| _{L^{2}}^{1-r}\|
f\| _{\dot{H}^{1}}^{r} \\
&\leq C\| f\| _{L^{2}}^{1-r}\| \nabla f\|_{L^{2}}^{r},
\end{align*}
and so the proof is complete.
\end{proof}

Since
$L^{1/3}(\mathbb{R}^3)\subset \dot{X}_r(\mathbb{R}^3)\mathbb{\subset }
\dot{\mathcal{M}}_{2,\frac{3}{r}}(\mathbb{R}^3)$,
the above regularity criterion alloy us to improve the results obtained
by Fan and Gao \cite{FG}.
Our main result on \eqref{eqNS} reads as follows.

\begin{theorem}\label{th1}
Let $\tilde{u}=u_{1}e_{1}+u_{2}e_{2}$ be the first two components
of a Leray-Hopf weak solution of the Navier-Stokes equation
corresponding to
$u_0\in H^{1}(\mathbb{R}^3)$ with $\mathop{\rm div}u_0=0$.
Suppose that
$\nabla \tilde{u}\in L^{\frac{2}{1-r}}(0,T,\dot{\mathcal{M}}_{2,3/r}
(\mathbb{R}^3))$ with $0<r<1$, then $u$ becomes the classical
solution on $(0,T] $.
\end{theorem}

\begin{proof}
We follow the ideas of the proof in \cite{FG}. By differentiating the
equations \eqref{eqNS} with respect to $x_{k}$, we take the scalar product
with $\partial _{k}u$, and integrate over $\mathbb{R}^3$. A
suitable integration by parts yields
\begin{equation}
\begin{aligned}
\frac{1}{2}\frac{d}{dt}\| \nabla u(t,.)\|
_{L^{2}}^{2}+\| \nabla ^{2}u(t,.)\| _{L^{2}}^{2}
&= -\int_{\mathbb{R}^3}\nabla [ (u.\nabla )u] .\nabla u\,dx
 \\
&= \sum_{i,j,k}\int_{\mathbb{R}^3}\partial _{k}u_{i}.\partial
_{i}u_{j}.\partial _{k}u_{j}dx.  \label{eq12}
\end{aligned}
\end{equation}
Following \cite{FG}, we only need to deal with the case $i=j=3$.
Since $\partial _{1}u_{1}+\partial _{2}u_{2}+\partial _{3}u_{3}=0$,
it follows that
\begin{align*}
\int_{\mathbb{R}^3}\partial _{k}u_{i}.\partial _{i}u_{j}.\partial
_{k}u_{j}dx
&= -\int_{\mathbb{R}^3}\partial _{k}u_{3}.(\partial
_{1}u_{1}+\partial _{2}u_{2}).\partial _{k}u_{3}dx \\
&\leq \int_{\mathbb{R}^3}| \nabla \widetilde{u}|
| \nabla u| ^{2}dx.
\end{align*}
Using H\"{o}lder's inequality and Lemma \ref{lem2}, we have
\begin{align*}
\int_{\mathbb{R}^3}| \nabla \widetilde{u}| |\nabla u| ^{2}dx
&\leq \| \nabla u\|
_{L^{2}}\| \nabla u\cdot \nabla \tilde{u}\| _{L^{2}}
\\
&\leq C\| \nabla \tilde{u}\| _{\dot{\mathcal{M}
}_{2,3/r}}\| \nabla u\| _{L^{2}}\| \nabla
u\| _{\dot{B}_{2,1}^{r}}   \\
&\leq C\| \nabla \tilde{u}\| _{\dot{\mathcal{M}
}_{2,3/r}}\| \nabla u\| _{L^{2}}\| \nabla
u\| _{L^{2}}^{1-r}\| \nabla ^{2}u\| _{L^{2}}^{r}
 \\
&= C\Big(\| \nabla \tilde{u}\| _{\dot{
\mathcal{M}}_{2,3/r}}^{\frac{2}{2-r}}\| \nabla u\|
_{L^{2}}^{2}\Big)^{\frac{2-r}{2}}\| \nabla ^{2}u\|
_{L^{2}}^{r}   \\
&\leq \frac{1}{2}\| \nabla ^{2}u\|
_{L^{2}}^{2}+C\| \nabla \tilde{u}\| _{\dot{
\mathcal{M}}_{2,3/r}}^{\frac{2}{2-r}}\| \nabla u\|
_{L^{2}}^{2}.
\end{align*} %\label{eq 13}
This estimates  combined with (\ref{eq12}), yield
\[
\frac{d}{dt}\| \nabla u(t,.)\| _{L^{2}}^{2}+\|
\nabla ^{2}u(t,.)\| _{L^{2}}^{2}\leq C\| \nabla \tilde{u}
\| _{\dot{\mathcal{M}}_{2,3/r}}^{\frac{2}{2-r}
}\| \nabla u\| _{L^{2}}^{2}.
\]
By Gronwall' s inequality we have
\[
\| \nabla u(t,.)\| _{L^{2}}^{2}\leq \| \nabla
u(0,.)\| _{L^{2}}^{2}\exp \Big(C\int_0^{T}\| \nabla
\tilde{u}(\cdot ,\tau )\| _{\dot{\mathcal{M}}_{2,\frac{3
}{r}}}^{\frac{2}{2-r}}d\tau \Big).
\]
This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
The author would like to express his gratitude to Professor Yong Zhou
for his valuable advice and interesting remarks.

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\end{document}